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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Rings of coinvariants and $ p$-subgroups


Author: Tzu-Chun Lin
Journal: Proc. Amer. Math. Soc. 138 (2010), 4243-4247
MSC (2000): Primary 13A50; Secondary 20F55
Published electronically: July 1, 2010
MathSciNet review: 2680050
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Abstract: Let $ \varrho :G\hookrightarrow GL(n, \mathbb{F})$ be a faithful representation of a finite group $ G$ over the field $ \mathbb{F}$ and let $ V \cong \mathbb{F}^n$ be an $ \mathbb{F}(G)$-module. It has been shown by L. Smith that if $ n=3$ and the order of $ G$ is divisible by the positive characteristic $ p$ of $ \mathbb{F}$, then $ \mathbb{F} [V]^G$ is Cohen-Macaulay. Under the condition $ n=3$ we prove the following conjecture through this remarkable result: If $ \mathbb{F} [V]_G$ is a Poincaré duality algebra, then $ \mathbb{F} [V]_{\operatorname{Syl}_{p}(G)}$ is a complete intersection, where $ \operatorname{Syl}_{p}(G)$ is a Sylow $ p$-subgroup of $ G$.


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Tzu-Chun Lin
Affiliation: Department of Applied Mathematics, Feng Chia University, 100 Wenhwa Road, Tai- chung 407, Taiwan, Republic of China
Email: lintc@fcu.edu.tw

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10470-7
PII: S 0002-9939(2010)10470-7
Keywords: Invariant theory, invariant polynomials, Gorenstein ring, Poincaré duality algebra, complete intersection
Received by editor(s): October 19, 2005
Received by editor(s) in revised form: May 13, 2008, October 20, 2008, and February 26, 2010
Published electronically: July 1, 2010
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.